Abstract

Although the spectra of drop radii in rain showers are broad, the supernumerary bows are caused by only those drops with radii of about 0.25 mm. The angle of minimum deviation, the rainbow angle, is a function of drop size, being large for big drops, owing to drop distortion, and large for small drops, owing to interference. Between these extremes, there is a minimum rainbow angle. The drops that cause it give rise to the supernumerary bows.

© 1983 Optical Society of America

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References

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  1. T. Young, “Experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. London1–16 (1804). This is the publication of the Bakerian Lecture of 1803.
  2. G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. VI, 397–403 (1838); Trans. Cambridge Philos. Soc. VIII, 595–600 (1849).
  3. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 245.
  4. P. Lenard, “Ueber Regen,” Meteorol. Zeit. 21, 248–262 (1904).
  5. J. E. McDonald. “The shape and aerodynamics of large raindrops,” J. Meteorol. 11, 478–494 (1954).
    [CrossRef]
  6. W. Moebius, “Zur Theorie des Regenbogens un ihrer experimentellen Prüfung,” Ph.D. Dissertation, Leipzig, 1907; Abh. Wiss. Kgl. Ges. Sachsen Math. Phys. Kl. (1907); also see F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” in The Physics of Precipitation, H. Weickmann, ed., Vol. 5 of American Geophysical Union Monograph Series (American Geophysical Union, Washington, D.C., 1960), pp. 280–286.
  7. H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, The Netherlands, 1978), p. 315.
  8. A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975), Eq. (6).
    [CrossRef]

1975 (1)

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975), Eq. (6).
[CrossRef]

1954 (1)

J. E. McDonald. “The shape and aerodynamics of large raindrops,” J. Meteorol. 11, 478–494 (1954).
[CrossRef]

1904 (1)

P. Lenard, “Ueber Regen,” Meteorol. Zeit. 21, 248–262 (1904).

1838 (1)

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. VI, 397–403 (1838); Trans. Cambridge Philos. Soc. VIII, 595–600 (1849).

1804 (1)

T. Young, “Experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. London1–16 (1804). This is the publication of the Bakerian Lecture of 1803.

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. VI, 397–403 (1838); Trans. Cambridge Philos. Soc. VIII, 595–600 (1849).

Green, A. W.

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975), Eq. (6).
[CrossRef]

Klett, J. D.

H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, The Netherlands, 1978), p. 315.

Lenard, P.

P. Lenard, “Ueber Regen,” Meteorol. Zeit. 21, 248–262 (1904).

McDonald, J. E.

J. E. McDonald. “The shape and aerodynamics of large raindrops,” J. Meteorol. 11, 478–494 (1954).
[CrossRef]

Moebius, W.

W. Moebius, “Zur Theorie des Regenbogens un ihrer experimentellen Prüfung,” Ph.D. Dissertation, Leipzig, 1907; Abh. Wiss. Kgl. Ges. Sachsen Math. Phys. Kl. (1907); also see F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” in The Physics of Precipitation, H. Weickmann, ed., Vol. 5 of American Geophysical Union Monograph Series (American Geophysical Union, Washington, D.C., 1960), pp. 280–286.

Pruppacher, H. R.

H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, The Netherlands, 1978), p. 315.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 245.

Young, T.

T. Young, “Experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. London1–16 (1804). This is the publication of the Bakerian Lecture of 1803.

J. Appl. Meteorol. (1)

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975), Eq. (6).
[CrossRef]

J. Meteorol. (1)

J. E. McDonald. “The shape and aerodynamics of large raindrops,” J. Meteorol. 11, 478–494 (1954).
[CrossRef]

Meteorol. Zeit. (1)

P. Lenard, “Ueber Regen,” Meteorol. Zeit. 21, 248–262 (1904).

Philos. Trans. R. Soc. London (1)

T. Young, “Experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. London1–16 (1804). This is the publication of the Bakerian Lecture of 1803.

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. VI, 397–403 (1838); Trans. Cambridge Philos. Soc. VIII, 595–600 (1849).

Other (3)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 245.

W. Moebius, “Zur Theorie des Regenbogens un ihrer experimentellen Prüfung,” Ph.D. Dissertation, Leipzig, 1907; Abh. Wiss. Kgl. Ges. Sachsen Math. Phys. Kl. (1907); also see F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” in The Physics of Precipitation, H. Weickmann, ed., Vol. 5 of American Geophysical Union Monograph Series (American Geophysical Union, Washington, D.C., 1960), pp. 280–286.

H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, The Netherlands, 1978), p. 315.

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Figures (7)

Fig. 1
Fig. 1

Light rays entering a spherical raindrop are deviated by 138 deg or more, so that rays striking near the edge of the drop will emerge in the same direction as rays that strike closer to the center of the drop.

Fig. 2
Fig. 2

The interference pattern, produced because the emerging wave front folds over on itself, is simulated with a moiré pattern. Large drops (a) produce closely spaced supernumerary bows, whereas small drops (b) produce widely spaced ones.

Fig. 3
Fig. 3

The angular deviation of the maximum brightness for the primary rainbow (lowest curve) and the first two supernumerary bows as given by Airy’s theory.

Fig. 4
Fig. 4

Ray tracing though an ellipsoidal drop shows that the minimum deviation is now greater that 138 deg.

Fig. 5
Fig. 5

The minimum deviation angle for ellipsoidal drops as a function of drop radius (of an equivalent volume sphere) and solar elevation.

Fig. 6
Fig. 6

The dashed lines are the three curves from Fig. 3. The dotted line is the 0-deg solar-elevation curve from Fig. 5. The solid curves show the position of the primary bow (lowest curve) and the first two supernumerary bows when both wave interference and drop ellipticity are taken into account. The V’s mark the drop radii that would cause the measurements extracted by Vaucher; the S’s are for those of Sardie. The overbar indicates the mean of the five measurements. Each datum point is plotted with an arbitrary ordinate (an angular separation was measured, not an angular deviation). Only the position along the abscissa is of interest.

Plate I
Plate I

(Alistair B. Fraser, p. 1626). The top of a rainbow, which shows two supernumerary bows. © Alistair B. Fraser.